TITLE: If all Subgroups are Cyclic, is group Cylic?
QUESTION [5 upvotes]: I am having difficulty seeing if it is the case that if all subgroups of a group are cyclic, that the group itself is cyclic.

REPLY [12 votes]: If you mean proper subgroups, then No! consider $G = \mathbf{Z}_2 \times \mathbf{Z}_2$.